Sorry if you know this already: but it sounds like you are looking still just at the STRs (the 37 markers and the 111 markers refer to this). But the big thing you get out of the Big-Y update is measurements of SNPs, which are a different sort of measurement than STRs.
STRs are repeated bits of DNA, and the numbers you see when you look at your results are the numbers of times the bits were measured to repeat in your DNA. So if a bit is "AC", and you had a count of 10, that means you had
ACACACACACACACACACAC
(there's 10 of those AC's there). When a mutation occurs, the number of repeats goes up or down. These kinds of mutations occur pretty frequently, as far as mutations go -- it doesn't take that many generations before you see a change in one of the numbers. This is both good and bad for genealogy. It is good because you can distinguish a lot of people in a recent time frame, because probably some mutations will have happened to separate more distant relations from closer ones. But these STRs are prone to "convergence", where one person's numbers move towards a distant relation's, by chance, and "back-mutation," where a mutation reverts back in a later generation. This makes it hard to build a certain tree of how lots of men are related.
SNPs are individual mutations at single places in your DNA. There are many, many more potential SNP positions in the Y-chromosome as well. So while a position will only very rarely mutate, there are so many positions that usually there will be an SNP mutation somewhere in the part of the Y-chromosome measured by BigY-700 within, say, three generations or so. So if there is a man in the same general part of the tree as you, and he shares an SNP mutation with you, then you can be just about certain that you both descend from the common patriline ancestor who first had that SNP. And if you have an idea from genealogical research who this man is, with plausible trails of records from both of you back to this man, then you can get much more certainty about your connection than if you are just, say, exact matches at 37 STR markers. And using the numbers of unique SNPs that you share, and also your STR measurements, the family trees of your matches, estimates of mutation rates, and probably other fancy stuff, FTDNA has this wonderful new algorithm to determine the approximate time that the ancestor lived. This is much more precise than the estimates we used to get just with STR measurements.
The reason you aren't exact matches at 111 markers anymore is because the 37 marker test is a subset of the 111 markers. While you had the same counts at all of the 37 STR markers in the smaller test, there were two differences in your counts at the 111 test. This still indicates you are pretty closely related. You are lucky to have these good matches!
I would definitely try to contact those people and see if you can convince them to upgrade to BigY. I've even paid for people to upgrade out of my own pocket before if they seem like the upgrade could be really informative.
